Lecture # 2 - Matrix Algebra

Transcription

1 Lecture # - Matrix Algebra Consider our simple macro model Y = C + I + G C = a + by Y C = I + G by + C = a This is an example of system of linear equations and variables In general, a system of x: a 11 x 1 + a 1 x = d 1 a 1 x 1 + a x = d Consider our model of supply and demand Q d = a bp demand equation Q s = c + dp : supply equation Q d = Q s Q d + bp + 0 = a demand equation 0 dp + Q s = c: supply equation Q d + 0 Q s = 0 1

2 And a system of m linear equations and n variables: a 11 x 1 + a 1 x + ::: + a 1n x n = d 1 a 1 x 1 + a x + ::: + a n x n = d :::::::::::::::::::::::::::::::::::::::::::::::::::::: a m1 x 1 + a m x + ::: + a mn x n = d m If there is at least a solution! consistent system If there is NO solution! inconsistent system There are three ingredients in that system of equations Set of coe cients a ij Set of variables x 1 ; x ; :::; x n Set of constant terms d 1 ; d ; :::; d m We can arrange them in three rectangular arrays: A = 6 4 a 11 a 1 ::: a 1n a 1 a ::: a n ::: ::: ::: ::: a m1 a m ::: a mn 7 x = 6 4 x 1 x. x n 7 d = 6 4 d 1 d. d n 7

3 Matrices De nition: A rectangular array of numbers, parameters or variables The number in the i-th row and j-th column (the number in position (i; j)) is denoted by a ij, i.e. A mn = fa ij g If a matrix has m n elements, it is of dimension m n Vector: special matrix. Row vector: Has dimension 1 n Column vector: Has dimension m 1 Other special matrices If m = n : square matrix (DEFINE main diagonal) Identity matrix (I): square matrix with 1 in main diagonal Formally: I has element (I) ii = 1 for all i = 1; :::; n; and (I) ij = 0 for i 6= j Null matrix: matrix N with (N) ij = 0 for all i = 1; :::; m, and j = 1; :::; n. Not necessarily a square matrix. Matrix algebra can be used: 1. To express the system of equations in a compact manner. To nd out whether solution to a system of equations exist. Need n linearly independent equations (where n is the number of elements in x). To obtain the solution if it exists. Note: Matrix linear algebra is applicable only to linear systems of equations

4 Examples: Consider our simple macro model Y = C + I + G C = a + by Y C = I + G by + C = a Matrix of coe cients A 1 1 A = b 1 Consider our model of supply and demand Q d = a bp demand equation Q s = c + dp : supply equation Q d = Q s Q d + bp + 0 = a demand equation 0 dp + Q s = c: supply equation Q d + 0 Q s = 0 Matrix of coe cients A 1 b 0 A = 4 0 d

5 Matrix Operations Matrices are important because they are easy to manipulate Equality: If A mn = fa ij g and B mn = fb ij g are two matrices of same dimension, they are equal if a ij = b ij for all i; j Addition and substraction If A mn = fa ij g and B mn = fb ij g have same dimension, then A + B implies a ij + b ij for all i; j So we add corresponding elements Same with substraction Give example Properties: A + B = B + A A + (B + C) = (A + B) + C Scalar multiplication Multiply each element of matrix A with the constant c Give example

6 Matrix multiplication Multiplication of matrices needs to satisfy a dimensional requirement: If we want A B; we need that the number of columns in A equals the number of rows in B So, if A mn = fa ij g and B pq = fb ij g ; for AB we need n = p If n = q; then the new matrix AB has dimensions m q To continue, consider an example: 1 A = 4 8 B 1 = De ne C = AB : matrix C will have dimension 1 : C = 4 To calculate the element of C; in row i and column j we nd the inner product of row i in A and row j in B Inner product: we take row i in A and row j in B; pair the elements sequentially, multiply each pair, and take the sum of the resulting product Example: to nd c 11 ; we take row 1 in A (1 ) and column 1 in B.( 9) pair each element sequentially: (1 with, and with 9) multiply each pair: (1)()=, ()(9)=7 take the sum of the resulting products: +7==c 11 In the same way, we can nd c 1 = a 1 b 11 + a b 1 = () () + (8) (9) = 8 and c 1 = 0: In general, to nd the value of element c ij we take row i in A and row j in B and n the inner product: c ij = c 11 c 1 c 1 nx a is b sj = a i1 b 1j + a i b j + ::: + a in b nj s=1 6

7 Examples: Consider our simple macro model Y = C + I + G C = a + by Y C = I + G by + C = a In matrix form Ax = d 1 1 A = x = b 1 Y C d = You can verify that Ax gives a matrix I + G a Y C by + C Consider our model of supply and demand Q d = a Q s = Q d = Q s bp demand equation c + dp : supply equation Q d + bp + 0 = a demand equation 0 dp + Q s = c: supply equation Q d + 0 Q s = 0 In matrix form Ax = d 1 b 0 A = 4 0 d 1 x = Q d P Q s d = 4 You can verify that Ax gives a matrix 4 a c 0 Q d + bp dp + Q s Q d Q s 7

8 Properties of multiplication No commutative law: AB 6= BA: Further, some times only one of them is de ned Exceptions B: is a scalar, the identity matrix I, or the inverse of A Associative law (AB)C = A(BC) Left distributive law: A(B + C) = AB + AC Right distributive law: (A + B)C = AC + BC but (A + B)C 6= C(A + B) (ca)b = A(cB) = cab If A mn ; then AI n = A; I m A = A For null matrices: A + N = N + A = A AN = N If A is a square matrix, we canwrite AA = A For an identity matrix of any dimension, II = I = I We cannot write A=B 8